HYACINTHOS 29424

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of H

 

Denote:

 

Na, Nb, Nc = the NPC centers of OBC, OCA, OAB, resp.

 

A'B'C', NaNbNc are 

1. Orthologic

2. Parallelogic.

 

GENERALIZATION

 

Let ABC be a triangle, P, Q two isogonal conjugate points and A'B'C' the pedal triangle of P.

 

Denote:

 

Na, Nb, Nc = the NPC centers of QBC, QCA, QAB, resp.

 

Which is the locus of P such that A'B'C', NaNbNc are:

1. Orthologic ?

2. Parallelogic?

3. Perspective?

 

 

[Ercole Suppa] 

 

Hi Antreas, 

 

1. Entire plane

 

2. Entire plane

 

3. The locus of P such that the A'B'C', NaNbNc are perspective: {q2 = circumcircle}∪{q5 = circumquintic through X(3), X(5)}

 

q2: c^2 x y + b^2 x z + a^2 y z = 0 (barys) 

 

q5: ∑ b^2 (b-c) c^2 (b+c) (7 a^6-19 a^4 b^2+17 a^2 b^4-5 b^6-19 a^4 c^2+2 a^2 b^2 c^2+5 b^4 c^2+17 a^2 c^4+5 b^2 c^4-5 c^6) x^3 y z+(a-b) (a+b) c^2 (a^8+9 a^6 b^2-20 a^4 b^4+9 a^2 b^6+b^8-7 a^6 c^2-16 a^4 b^2 c^2-16 a^2 b^4 c^2-7 b^6 c^2+15 a^4 c^4+20 a^2 b^2 c^4+15 b^4 c^4-13 a^2 c^6-13 b^2 c^6+4 c^8) x^2 y^2 z-a^4 c^2 (2 a^6-5 a^4 b^2+4 a^2 b^4-b^6-6 a^4 c^2+a^2 b^2 c^2-b^4 c^2+6 a^2 c^4+4 b^2 c^4-2 c^6) y^4 z+a^4 (2 a^8-7 a^6 b^2+9 a^4 b^4-5 a^2 b^6+b^8-3 a^6 c^2+5 a^4 b^2 c^2-6 a^2 b^4 c^2+4 b^6 c^2-3 a^4 c^4-7 a^2 b^2 c^4-11 b^4 c^4+7 a^2 c^6+9 b^2 c^6-3 c^8) y^3 z^2-a^4 (2 a^8-3 a^6 b^2-3 a^4 b^4+7 a^2 b^6-3 b^8-7 a^6 c^2+5 a^4 b^2 c^2-7 a^2 b^4 c^2+9 b^6 c^2+9 a^4 c^4-6 a^2 b^2 c^4-11 b^4 c^4-5 a^2 c^6+4 b^2 c^6+c^8) y^2 z^3+a^4 b^2 (2 a^6-6 a^4 b^2+6 a^2 b^4-2 b^6-5 a^4 c^2+a^2 b^2 c^2+4 b^4 c^2+4 a^2 c^4-b^2 c^4-c^6) y z^4 = 0 (barys)

 

Let P(x:y:z) (barys) and Q = Q(P) the perspector of A'B'C' and NaNbNc.

 

*** Pairs {P = X(i) ∈ q2, Q = X(j)} for these {i,j}: {98, 13137}, {99, 12833}, {110, 7471}

 

*** Pairs {P = X(i) ∈ q5, Q = X(j)} for these {i,j}: {3,5}

 

*** Some points:

 

Q(X(5)) = X(54)X(143) ∩ X(546)X(6346)

 

= a^2 (a^4-2 a^2 b^2+b^4-2 a^2 c^2-3 b^2 c^2+c^4) (2 a^10-5 a^8 b^2+2 a^6 b^4+4 a^4 b^6-4 a^2 b^8+b^10-5 a^8 c^2+2 a^6 b^2 c^2+a^4 b^4 c^2+5 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4+a^4 b^2 c^4-2 a^2 b^4 c^4+2 b^6 c^4+4 a^4 c^6+5 a^2 b^2 c^6+2 b^4 c^6-4 a^2 c^8-3 b^2 c^8+c^10) : : (barys)

 

= (15 R^4+19 R^2 SB+19 R^2 SC+3 R^2 SW-6 SB SW-6 SC SW-2 SW^2)S^2-R^4 SB SC+R^2 SB SC SW+2 SB SC SW^2 : : (barys)

 

= lies on these lines: {54,143}, {546,6346}, {1209,10272}, {9969,15516}, {10095,11817}, {10610,11557}, {11561,12041}, {13367,14449}, {13491,33541}

 

= (6-9-13)  search numbers:  [0.0812798866664975914, -0.3894250816532193027, 3.8727526676674551166]

 

------------------------------------------------------------------------------

 

Q(X(74)) = ISOGONAL CONJUGATE OF X(15469)

 

= (a^4-2 a^2 b^2+b^4+a^2 c^2+b^2 c^2-2 c^4) (a^4+a^2 b^2-2 b^4-2 a^2 c^2+b^2 c^2+c^4) (2 a^8-2 a^6 b^2-a^4 b^4+b^8-2 a^6 c^2+4 a^4 b^2 c^2-4 b^6 c^2-a^4 c^4+6 b^4 c^4-4 b^2 c^6+c^8) : : (barys)

 

= S^4 + (-18 R^2 SB-18 R^2 SC-3 SB SC+12 R^2 SW+4 SB SW+4 SC SW-3 SW^2)S^2 -324 R^4 SB SC+144 R^2 SB SC SW-15 SB SC SW^2 : : (barys)

 

= 2*X[113]-X[14611], 2*X[140]-3*X[21315], X[477]-2*X[3154], 2*X[548]-X[21317], 2*X[3233]-X[12383], X[3258]-2*X[7687], X[10295]-2*X[11657], 3*X[10706]+X[31874], 2*X[11801]-X[16340], 4*X[12068]-3*X[15035], 4*X[12900]-3*X[31378], X[16163]-2*X[22104], 3*X[23515]-2*X[31379]

 

= lies on the curves K025, Q092, Q106 and these these lines: {4,523}, {5,14385}, {30,74}, {113,14611}, {140,21315}, {230,32640}, {316,1494}, {381,9717}, {403,1300}, {477,3154}, {542,1553}, {546,3470}, {548,21317}, {671,9139}, {1263,11558}, {2777,6070}, {3233,12383}, {3258,7687}, {5523,8749}, {5962,10152}, {7471,15468}, {10113,16168}, {10295,11657}, {10297,14919}, {10706,31874}, {11801,16340}, {12068,15035}, {12900,31378}, {13202,32417}, {16163,22104}, {16243,25338}, {22265,32111}, {23515,31379}

 

= isogonal conjugate of X(15469)

 

= antigonal image of X(7471)

 

= midpoint of X(74) and X(14989)

 

= reflection of X(i) in X(j) for these {i,j}: {5,21316}, {74,12079}, {477,3154}, {3258,7687}, {7471,25641}, {10295,11657}, {12383,3233}, {14611,113}, {14934,5}, {16163,22104}, {16340,11801}

 

= {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {74,5627,12079}, {477,14644,3154}, {5627,14989,74}

 

= (6-9-13)  search numbers:  [-6.1436603840234256067, -6.0605851061983525639, 10.6719897326709734659]

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Q(X(100)) = ANTICOMPLEMENT OF X(14115)

 

= a (a-b) (a-c) (a^4 b^2-2 a^2 b^4+b^6-2 a^3 b^2 c+2 a^2 b^3 c+2 a b^4 c-2 b^5 c+a^4 c^2-2 a^3 b c^2+2 a^2 b^2 c^2-2 a b^3 c^2-b^4 c^2+2 a^2 b c^3-2 a b^2 c^3+4 b^3 c^3-2 a^2 c^4+2 a b c^4-b^2 c^4-2 b c^5+c^6) :: (barys)

 

= 3*X[2]-2*X[14115], X[3025]-2*X[3035], X[3937]-2*X[22102], 3*X[10707]+X[31877], 5*X[31272]-4*X[33646]}

 

= lies on these lines: {2,14115}, {4,8}, {59,108}, {100,513}, {912,18341}, {1331,2222}, {2810,6075}, {2818,6073}, {3025,3035}, {3937,22102}, {3952,20293}, {5375,14298}, {5854,13756}, {10707,31877}, {11681,31849}, {15313,15343}, {31272,33646}

 

= reflection of X(i) in X(j) for these {i,j}: {{100,15632}, {3025,3035}, {3937,22102}}

 

= anticomplement of X(14115) 

 

= (6-9-13)  search numbers:  [1.9063676375075504026, -0.2696936351310823761, 2.9475134735331821008]

 

Best regards

Ercole Suppa